Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Rademacher theorem for manifolds

Let $M$ be a smooth manifold. Let $d$ be any metric on $M$ which generates the topology of $M$. Let $f:M \to R$ be Lipschitz w.r.t the metric $d$. Is it true that $f$ is differentiable a.e? Note: ...
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The second differential as a differential on the double tangent bundle

I know what the second differential of $f : \Bbb R^n \to \Bbb R$ means. Nevertheless, when working with abstract manifolds and in the absence of a connection, one cannot come up with a 2-covariant ...
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28 views

First and second fundamental form with rotational surfaces (check)

I'm working out some examples for surfaces in differential geometry. I was working out simple rotational surface, but I think I've done something wrong. Let $\gamma\left(t\right)$ a curve ...
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Product-like metric on a pseudo-Riemmanian manifold foliated by Lie group orbits

Suppose we have an $n$-dimensional pseudo-Riemmanian manifold $(M,g)$ on which a connected Lie group $G$ acts isometrically (I am most interested in the Lorentzian case if it matters). Suppose that ...
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29 views

Is there a relationship between the pullback in differential geometry and that in category theory?

1. Is there a relationship between the pullback in differential geometry and the pullback in category theory? [2. Is there a relationship between the pushforward/pushout in differential geometry ...
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What is the geometrical meaning of the integral of a vector valued function?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is an integrable function. then $\int_a^b f(x)dx$ can be considered as the area between the graph and the x-axis. But what if $f:\mathbb{R}^n\rightarrow \...
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Interpretation of hybrid planar / spherical volume element

I have a volume element and I would like to know if anyone recognizes it as coming naturally from any kind of embedding or projection or something like that. The volume element is: $d^3V = (d\chi)(\...
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83 views

Ham Sandwich Theorem - intuitive proof

Ham Sandwich Theorem. Given 3 measurable "objects" in $\mathbb{R}^3$, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single 2-dimensional plane. Can ...
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36 views

Conformal class of $\mathbb S^n$ [on hold]

What can we say about the conformal class of the sphere $\mathbb S^n$?
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33 views

Does a differential manifold implies existence of unique tangent space at every point?

I like differential geometry and I want to know if a differentiable manifold implies unique tangent space at every point. I have searched but the definition I have found of differential manifold is ...
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29 views

How do you find the metric tensor for a given manifold?

Is there some general way to derive the metric tensor for a given manifold M? For example, how was the metric for the surface of a sphere $$ds^2=d\theta^2+\sin^2\theta \, d\phi^2$$ first derived?
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Total Gaussian curvature of a flat surface with cone singularities

Let $S$ be a surface and $g$ a riemannian metric on $S$ which is flat with finite number of isolated conical singularities of cone angle $\theta_i>2\pi$. I have two questions: 1) of course the ...
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36 views

How to distinguish between arc length and arc length parametrisation?

I am trying to understand and distinguish the difference between arc length and arc length parameterisation. The first thing, how do denote the $\text{arc length}$ and $\textit{arc length ...
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83 views
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Prove that $g$ is a submanifold: $g (t,u,v) = (t^2,u^2,v^2,\sqrt{2}uv, \sqrt{2}tv,\sqrt{2}tu)$

We consider $g : (t,u,v)\in \mathbb{R}^3 \mapsto (t^2,u^2,v^2,\sqrt{2}uv, \sqrt{2}tv,\sqrt{2}tu)\in\mathbb{R}^6$. I have to prove that $g(\mathbb{S}^2)$ is a submanifold of $\mathbb{R}^6$. $dg_{(t,u,...
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Can I choose this kind of neighborhood of a point on a curve?

$\textbf{Question}$ Suppose $f:[0,1]\rightarrow\mathbf{R}^{2}$ is a continuously differentiable, 1-1 function. If $f(a)\in f([0,1])$, then should there be some open ball $B(f(a),\epsilon)$, centered ...
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24 views

What is the relationship between line of curvature and umbilical point?

I am guessing whether or not the following statement is true: All the points lie on a line of curvature of a connected curve are umbilical points. Conversely, given an umbilical point on a surface, ...
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1answer
27 views

How to know whether a contact form is only defined locally or globally?

As described e.g. here the following is the standard contact form on $\mathbb R^{2n+1}$: $$ \omega = dz + \sum_{k=1}^n x_k dy_k$$ Similarly, the following is the standard contact form on $S^{2n+1}$: ...
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Convergence of a integral for every curve in the sphere

Let $S$ be the unit open sphere in $\mathbb{R}^3$: $x^2+y^2+z^2< 1$ and $\partial S$ its border $x^2+y^2+z^2= 1$. Let $f:S\cup \partial S\rightarrow \mathbb{R}$ be a continuous function which is ...
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32 views

Proving this connection is actually the Levi-Civita connection.

Having a manifold $M$ and some vector bundle $E$ over $M$ I am familiar with the definition of a connection given by a function $\nabla:\chi(M)\times\Gamma(E)\rightarrow \Gamma(E)$ that satisfies the ...
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2answers
106 views

Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...
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Foci of an ellipse and focal point of a manifold

Given an embedded $k$-manifold $M\subset\mathbb{R}^n$, a focal point $e\in \mathbb{R}^n$ at $q\in M$ is defined as $e=q+v$, where $(q,v)$ is a critical point of the map $$\begin{align*}E:\nu(M)&\...
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19 views

A natural Poisson bivector on the tangent bundle?

For a smooth manifold $M$, there is a natural $1$-form $\theta$ on $T^*M$ such that $\Bbb d \theta$ is a symplectic form. Somewhat symetrically, on $TM$ there is a natural tangent field $V$. Is it ...
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90 views

Connection on Submanifold using given connection on ambient manifold

This result features in Hicks' Notes on Differential Geometry book.The theorem states that given $C^\infty$ fields X and Y on a submanifold M, we have $$\bar D_X Y=D_X Y+V(X,Y)$$ where $\bar D$ is a ...
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are immersions of surfaces in $\mathbb R^3$ dense in all regular maps?

Let $u\in C^\infty(\Omega,\mathbf R^3)$ with $\Omega$ open set in $\mathbf R^2$. Can we find $u_k\in C^\infty(\Omega,\mathbf R^3)$ with $\mathrm{rank}(Du_k(x))=2$ for all $x\in\Omega$ such that $u_k \...
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Derivation of a representation through a vector field

Question: (Exercise 3.4.12 - Sharpe) Let $H$ be a Lie group, $V$ a vector space, and $\rho: H \to Gl(V)$ a representation. Let $U$ be a manifold, $X$ a vector field on $U$, and $h: U \to H$ and $f:...
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when do we say a parametrized curve's orientation is consistent with an oriented plane curve

Given an oriented plane curve $C$ and a point $p$ on $C$.Let $A:I\to C$ be a parametrization of a segment of $C$ which contains $p$, where $I \subseteq R$. when do we say $A$ is oriented consistently ...
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1-parametric subgroups of diffeomorphisms induce a complete vector field

I have been working through this book on differential equations and I do not quite understand the justification for one claim. Namely, the author claims that every 1-parameter subgroup $\{\psi_\...
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1answer
87 views

Why are geodesically convex sets diffeomorphic to $\Bbb R^n$?

In the construction of a good cover for a manifold $M^n$, Bott & Tu use the fact that each point in $M$ is contained in a geodesically convex set (after picking a Riemannian metric). They then ...
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Equivalence of definitions of harmonic (or wave) coordinates

In GR, one often uses harmonic (or wave) coordinates to simplify things. Now, one definition involves the coordinates themselves: $$ \Box_g x^{\alpha} = 0 $$ where $ \Box_g = g_{\mu \nu}\nabla^{\mu}...
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1answer
29 views

Integral of $\omega\wedge\overline{\omega}$ on Riemann surface

Let $X$ be a Riemann surface of genus $g$ and $\omega$ a meromorphic 1-form on it. I've read that if $\omega$ has just a simple pole in $x\in X$ (and is holomorphic on $X\setminus\{x\}$) then the ...
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88 views

About codifying length and energy using a one-form

Let $M^n$ be a smooth manifold and $g$ be a Riemannian metric on $M$. Is there $\omega \in \Omega^1(M)$ such that $\int_c \omega = \ell(c)$ for every curve in $M$? In general the answer seems ...
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2answers
31 views

Hyperbolic metric geodesically complete

Consider the upper half plane model of the hyperbolic space ($\mathbb{H}$ with the riemannian metric $g=\frac{dx^2+dy^2}{y^2}$). It is known that $(\mathbb{H},g)$ is geodesically complete, which means ...
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1answer
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Viewing complex projective space as a Grassmannian manifold

Complex projective space $\mathbb{CP}^n$ carries the structure of a complex manifold of dimension $n$, hence has the underlying structure of a real manifold of dimension $2n$. It is the set of complex ...
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20 views

Can a section of a signed distance filed uniquely determine this field function?

A Signed distance field function is a field function which tells the minimum distance from any point in space to a specific object. Let $\phi(\vec{x})$ be a signed distance field function, an ...
2
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1answer
30 views

“Approximate Isometry” in Riemannian Geometry

I apologize if the notion I'm asking about is well known, I'm no expert in geometry (and I did not find an answer via google). Suppose $(X,g_X)$ and $(Y,g_Y)$ are (smooth) Riemannian manifolds. I'm ...
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66 views

Uniqueness of minimizing geodesic $\Rightarrow$ uniqueness of connecting geodesic?

Let $M$ be a complete connected Riemannian manifold. Fix $p \in M$. Assume every point in $M$ has a unique minimizing geodesic connecting it to $p$. Is it true that for every point, the only ...
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1answer
892 views

Relationships between curvature, torsion, unit tangent vector, and binormal vector of a curve

Homework has already been collected and graded (but no explanation given) for these problems. I'm curious how to approach the problem. Assume that the vector space we're in is $\Re^{3}$. Prove that ...
25
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2answers
278 views

Origins of Differential Geometry and the Notion of Manifold

The title can potentially lend itself to a very broad discussion, so I'll try to narrow this post down to a few specific questions. I've been studying differential geometry and manifold theory a ...
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34 views

Distributions on submanifolds

I am beginner in differential geometry. I stuck with the concept of distributions(like invariant, anti invariant, slant) on submanifolds. Can you explain what are distributions on submanifolds? If ...
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1answer
46 views

Neglected constant curvature difference surfaces

What are some surfaces where $ \kappa_1-\kappa_2$ is constant? On a sphere where all are umbilical points.. is a special case. For the $ \kappa_1+\kappa_2$ = constant case we have DeLaunay and ...
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1answer
11 views

Sectional curvature of 3-hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{\bigl\langle R(X,Y)X,Y\bigr\rangle}{|X|^2 |Y|^2 - \langle X,Y\...
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1answer
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metric and homotopic maps on a manifold

Let $Y\subset \mathbb{R}^n$ be an embedded manifold without boundary. Prove that there is $\epsilon>0$ with the following property: If $f,g \colon X \rightarrow Y$ are smooth maps defined on a ...
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Maximal offset distance for a surface

Let $\vec r = \vec r(u, v)$ be a regular (analytic) surface. Now we offsetting this surface to distance $d$ in normal direction; new surface is $\vec r' = \vec r + d\vec n$. New surface $\vec r'$ is ...
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1answer
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function from a genus $2$ surface to $S^1$

Let $f\colon \Sigma \rightarrow S^1$ be a map from a genus $2$ surface to $S^1$. If $y\in S^1$ is a regular value of $f$ and $f^{-1}(y)$ is a nonseparating circle of $\Sigma$. How can I prove that $f$...
2
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2answers
149 views

Interior product and exterior product

I have seen around the internet that this should hold: $$\iota_X(\alpha\wedge\beta)=\iota_X\alpha\wedge\beta+(-1)^k\alpha\wedge\iota_X\beta,$$ where $X$ is a vector field, $\alpha$ a $k$-form, $\...
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3answers
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Why the Jacobian isnt always 1? [on hold]

We have $A=\iint {\rm dx}' {\rm d}y'=\iint G \,{\rm d}x\,{\rm d}y$, where the integral is over a region with area $A$ in the $xy$-plane and $G$ the Jacobian of the coordinate transformation $x\to x'$ ...
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0answers
17 views

Effect of gauge transformation on connection 1-form of a principal connection

Let $(P,\pi,M,G)$ be a principal fibration, $A$ a principal connection on $P$ (i.e. $\forall p \in P, T_pP = A_p \oplus V_p$), $\omega$ the connection 1-form of $A$, $f$ a gauge transformation of $P$, ...
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Compactly supported form on $\mathbb{R}^n$

Let $\omega$ be a compactly supported smooth $n$-form on $\mathbb{R}^n$. Show that there exists a compactly supported smooth $(n-1)$-form $\eta$ with $\omega=d\eta$ if and only if $$ \int_{\mathbb{R}^...
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27 views

sectional curvature of hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{<R(X,Y)X,Y>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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1answer
22 views

Is the set where the exponential map is defined an open subset of $TM$?

Let $M$ be a connected Riemannian manifold. Define $O=\{(p,v) \in TM|\, \,exp_p(v) \text{ is defined} \}$. Is $O$ an open subset of $TM$? I know that for every point in $M$, there is a neighbourhood $...